• H test for a single population
• 1- and 2-sided tests
• H test for two populations
• Assumptions of parametric tests

• T-tests are parametric tests

• Parametric tests: specify/assume probability distribution from which parameters came

• Non-parametric tests: no assumption about probability distribution

• Mukasa et al 2021 DOI: 10.4236/ojbm.2021.93081

• If assumptions of parametric test violated, test becomes unreliable
• This is because test statistic may no longer follow distribution
• Most parametric tests robust to mild/moderate violations of below assumptions

• Basic assumptions of parametric t-tests:
• Normality, equal variance, random sampling, no outliers
• Normality: Samples from normally distributed population
  • Graphical tests: histograms, dotplots, boxplots, qq-plots
  • “Formal” tests: Shapiro-Wilk test

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• Equal variance: samples are from populations with similar degree of variability
• Graphical tests: boxplots
• “Formal” tests: F-ratio test
• Parametric tests most robust to violations of normality and equal var. assumptions when samples sizes equal

• Normality, equal variance, random sampling, no outliers
• Random sampling: samples are randomly collected from populations; part of experimental design
• Necessary for sample -> population inference

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• Normality, equal variance, random sampling, no outliers
• No outliers: no “extreme” values that are very different from rest of sample
• Graphical tests: boxplots, histograms
• “Formal tests”: Grubb’s test
• Note: outliers also problem for non-parametric tests

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• Perform 2-sample t-test:
  • pop A: 5.3, 5.6, 4.3, 4.9, 5.3, 4.1, 5.2, 5.0 cm
  • pop B: 6.1, 4.7, 5.9, 4.7, 6.2, 6.0, 5.4, 4.9 cm
• ȳ_popA - ȳ_popB = -0.53
• s_ȳopA-ȳopB = 0.29
• t = -1.80
• df = 14
• p (estimated from t-table) = 0.05 < p < 0.1
• Writeup: “a 2-tailed, independent 2-sampe t-test showed no significant difference bw beak length of pop A (4.96 cm ± 0.52 SD) and pop B (5.49 ± 0.64) at á=0.05: t(14) =-1.80, p = 0.094”

• Look through the ecological literature and find an example of a published manuscript that uses either a t-test of one of the tests mentioned in Q2. Provide the following information:
  • Reference for paper
  • Scientific question being addressed
  • Specific hypothesis tested (in mathematical notation)
  • The results of the t test (t, df, p) and the author’s conclusions

• Statistical significance: difference unlikely due to chance
• Says nothing about biological significance of difference!
• With large sample size can detect very small differences between populations
• E.g.: consider 2 snail populations, A and B: Ho: µ~size A~ = µ~size B~; Ha: µ~size A~ ≠ µ~size B~

• Size of A: 5.05 (± 2.00 SD)mm, size of B: 5.00 (± 2.00 SD)mm
• Sample 50, 200, 30,000 individuals from each pop:
  • n = 50: t = 0.32, df = 98, p-value = 0.75
  • n = 200: t = 0.058, df = 398, p-value = 0.95
  • n = 30,000: t = -4.47, df = 59998, p-value = 7.996*10^-6

• Finally, statistically significant difference…
• Meaningful? Ecologically significant? Statistics can’t answer this question
• IMPORTANT to report info that can assess biological significance
• “A two-tailed, two-sample independent t-test showed significant difference in size between pop. A (4.99 mm ± 1.99 SD) and pop. B (5.06 mm ± 1.99 SD) at á=0.05 (t = -4.47, df = 59998, p-value < 0.0001).”

• Basic assumptions of parametric t-tests:
• Normality, equal variance, random sampling, no outliers
• What to do if assumptions are violated?

• t-tests have several assumptions. Alternative tests, with more relaxed assumptions, are available to statisticians. In which case would you use the following tests?
  • Welch’s t-test: when distribution normal but variance unequal
  • Permutation test for two samples: when distribution not normal (but both groups should still have similar distributions and ~equal variance)
  • Mann-Whitney-Wilcoxon test: when distribution not normal and/or outliers are present (but both groups should still have similar distributions and ~equal variance)

• QQ-plots: tool for assessing normality
• On x- theoretical quantiles from SND
• On y- ordered sample values
• Deviation from normal can be detected as deviation from straight line

• In some cases, data can be mathematically “transformed” to meet assumptions of parametric tests

• Welch’s t-test: common “robust” test for means of two populations
• Robust to violation of equal variance assumption, deals better with unequal sample size
• Parametric test (assumes normal distribution)
• Calculates a t statistic but recalculates df based on samples sizes and s

• In R: t.test(y1, y2, var.equal = FALSE, paired = FALSE) will use the Welch approach

• Rank-based tests: no assumptions about distribution (non-parametric)
• Ranks of data: observations assigned ranks, sums (and signs for paired tests) of ranks for groups comparted
• Mann-Whitney U test common alternative to independent samples t-test
• Wilcoxon signed-rank test is alternative to paired t-test

• Assumptions: similar distributions for groups, equal variance
• Less power than parametric tests
• Best when normality assumption can not be met by transformation (weird distribution) or large outliers

• Permutation tests based on resampling: reshuffling of original data
• Resampling allows parameter estimation when distribution unknown, including SEs and CIs of statistics (means, medians)
• Common approach is bootstrap: resample sample with replacement many times, recalculate sample stats

• Sample A: n = 40, ȳ= 1.72, s = 4.17
• Sample B: n = 35, ȳ= 4.50, s = 4.83
• Ho: µ_A = µ_B, Ha: µ_A ≠µ_B
• Calculate ∆ in means bw two groups (2.78)

• Randomly reshuffle observations bw groups (keeping n_A=40 and n_B=35), calculate ∆
• Repeat >1,000 times
• Record proportion of the ∆means is ≥2.94 µmol
• This is equivalent to p-value and can be used in “traditional” H test framework
• For a graphical explanation: https://www.jwilber.me/permutationtest/

• In R (using ‘perm’ package):
• permTS(y1, y2, alternative = “two.sided”, method = “exact.mc”, control = permControl(nmc = 10000))
• Assumptions: both groups have similar distribution; equal variance

• Get practice doing basic t-tests
• Alternatives in next lecture
• Dataset (squirrel_data.csv) and lab instructions on Canvas
• Answer questions in bold
• Due end of Thursday